Related papers: Dunkl--Williams inequality for operators \\ associ…

In this paper, we compute universal inequalities of eigenvalues of a large class of second-order elliptic differential operators in divergence form, that includes, e.g., the Laplace and Cheng-Yau operators, on a bounded domain in a complete…

In this paper we prove inversion formulas for the Dunkl intertwining operator $V_k$ and for its dual ${}^tV_k$ and we deduce the expression of the representing distributions of the inverse operators $V_k^{-1}$ and ${}^tV_k^{-1}$, and we…

Several refinements of norm and numerical radius inequalities of bounded linear operators on a complex Hilbert space are given. In particular, we show that if $A$ is a bounded linear operator on a complex Hilbert space, then $$…

We give necessary and sufficient conditions in order that inequalities of the type $$ \| T_K f\|_{L^q(d\mu)}\leq C \|f\|_{L^p(d\sigma)}, \qquad f \in L^p(d\sigma), $$ hold for a class of integral operators $T_K f(x) = \int_{R^n} K(x, y)…

Let $n \geq 2$, let $\Omega \subset \mathbf{R}^n$ be a bounded domain with smooth boundary, and let $1 \leq p \leq 2$. We prove a reverse-Holder inequality for functions $u$ realizing the best constant in the Sobolev inequality, that is…

For bounded linear operators $A,B$ on a Hilbert space $\mathcal{H}$ we show the validity of the estimate $$ \sum_{\lambda \in \sigma_d (B)} \dist(\lambda, \overline{\num}(A))^p \leq \| B-A \|_{\mathcal{S}_p}^p$$ and apply it to recover and…

We provide a number of sharp inequalities involving the usual operator norms of Hilbert space operators and powers of the numerical radii. Based on the traditional convexity inequalities for nonnegative real numbers and some generalize…

We establish, for $1 < p < \infty$, higher order $\mathcal{S}^p$-differentiability results of the function $\varphi : t\in \mathbb{R} \mapsto f(A+tK) - f(A)$ for selfadjoint operators $A$ and $K$ on a separable Hilbert space $\mathcal{H}$…

For an unbounded self-adjoint operator D on a Hilbert space H and a bounded operator a on H we say that a is weakly D-differentiable if for any pair of vectors x, y in H the function <exp(itD) a exp(-itD)x, y> is differentiable at t =0. We…

We will show that for any two bounded linear operators $X,Y$ on a Hilbert space ${\frak H}$, if they satisfy the triangle equality $|X+Y|=|X|+|Y|$, there exists a partial isometry $U$ on ${\frak H}$ such that $X=U|X|$ and $Y=U|Y|$. This is…

Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces such that their intersection contains a fixed vector space D. It is of interest to make a precise…

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>4m-1$, $m\in \mathbb N$. We show that for any $\frac{2n}{n-4m+1}<p\leq \infty$ and $0\leq \alpha…

Let $P_+$ be the Riesz's projection operator and let $P_-= I - P_+$. We consider the inequalities of the following form $$ \|f\|_{L^p(\mathbb{T})}\leq B_{p,s}\|( |P_ + f | ^s + |P_- f |^s) ^{\frac 1s}\|_{L^p (\mathbb{T})} $$ and prove them…

The main objective of this paper is to present Ostrowski's inequality for a broader class of functions and to propose a refinement to the classical version of it. The original Ostrowski's inequality can be stated as follows "If…

In this expository article we collect and discuss some recent results on different consequences of a Sharp Reverse H\"older Inequality for $A_{\infty}$ weights. For two given operators $T$ and $S$, we study $L^p(w)$ bounds of…

Given a general symmetric elliptic operator $$ L\_{a} := \sum\_{k,,j=1}^d \p\_k (a\_{kj} \p\_j) + \sum\_{k=1}^d a\_k \p\_k - \p\_k(\overline{a\_k} .) + a\_0$$we define the associated Dirichlet-to-Neumann (D-t-N) operator with partial data,…

Let $H=-\Delta+V$ be a Schr\"odinger operator on $L^2(\mathbb R^n)$ with real-valued potential $V$ for $n > 4$ and let $H_0=-\Delta$. If $V$ decays sufficiently, the wave operators $W_{\pm}=s-\lim_{t\to \pm\infty} e^{itH}e^{-itH_0}$ are…

We establish inequalities that compare the p-Wasserstein distance to distances which are built as suprema of box measures. More precisely, when the measures are supported on $[0,1]^d$, we obtain sharp upper-bounds of the $p$-Wasserstein…

We prove variation and oscillation $L^p$-inequalities associated with fractional derivatives of certain semigroups of operators and with the family of truncations of Riesz transforms in the inverse Gaussian setting. We also study these…

We compute arithmetic support of the formal deformations $D=P+tQ_1+t^2Q_2+...$ of the differential operator $P=(x\partial_x-r_1)...(x\partial_x-r_k)$, where $r_1,...,r_k\in\mathbb{Q}$ for sufficiently large primes $p$ in terms of the…